AP Calculus BC Flashcards: Radius and Interval of Convergence of Power Series
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: July 2026
Review key ideas with interactive flashcards. This set includes 16 cards to help you master important concepts.
What is the primary goal when asked about the convergence of a power series?
The primary goal is to determine the radius of convergence and the full interval of convergence for the series.
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What is the primary goal when asked about the convergence of a power series?
The primary goal is to determine the radius of convergence and the full interval of convergence for the series.
What are the two possible outcomes for the convergence of a power series?
If a power series converges, it either converges at a single point or it converges over an interval of convergence.
If you determine that a power series centered at x=1 has a radius of convergence R=3, what is the open interval of convergence?
The open interval is $(1-3, 1+3)$, which is $(-2, 4)$. The endpoints must still be tested to find the full interval of convergence.
In the power series $\sum_{n=0}^{\infty} a_n (x-r)^n$, what does the value 'r' represent?
The value 'r' is the center of the power series and the center of its interval of convergence.
After finding the radius of convergence, what crucial step is needed to find the complete interval of convergence?
It is necessary to test both endpoints of the open interval to determine if the series converges or diverges at those specific points.
What is the radius of convergence?
The radius of convergence is a value that identifies an open interval on which a power series converges.
What does it mean if a power series converges only at a single point?
This means the radius of convergence is R=0, and the series only converges at its center, x=r.
If a power series has a positive radius of convergence, what is its relationship to a Taylor series?
The power series is the Taylor series of the function to which it converges over the open interval of convergence.
What is the interval of convergence?
The interval of convergence is the complete set of x-values for which a power series converges, including any endpoints where it converges.
A power series has a radius of convergence R=5. If you integrate the series, what is the new radius of convergence?
The new radius of convergence is also R=5, as it is the same as the radius of convergence of the original power series.
A power series is differentiated term-by-term. How does its new interval of convergence relate to the original?
The new series will have the same radius of convergence, but the convergence at the endpoints may have changed and must be re-tested.
Why is the radius of convergence not enough to define the full interval of convergence?
The radius of convergence only defines an open interval; the behavior at the endpoints must be tested separately to determine the complete interval.
What is a power series?
A power series is a series of the form $\sum_{n=0}^{\infty} a_n (x-r)^n$, where $r$ is a real number and $\{a_n\}$ is a sequence of real numbers.
How does term-by-term differentiation affect a power series's radius of convergence?
The radius of convergence of a power series obtained by term-by-term differentiation is the same as the radius of convergence of the original series.
How does term-by-term integration affect a power series's radius of convergence?
The radius of convergence of a power series obtained by term-by-term integration is the same as the radius of convergence of the original power series.
What test is commonly used to determine the radius of convergence of a power series?
The ratio test can be used to determine the radius of convergence of a power series.